THIN FILMS FLOW DRIVEN BY GRAVITY AND A SURFACE TENSION GRADIENT

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1 THIN FILMS FLO DRIVN BY GRAVITY AND A SURFAC TNSION GRADINT MILIA BORŞA The flow of a thin layer on a horizontal plate in the lbriation approximation i onidered. The flow i driven imltaneoly by gravity and ome gradient of rfae tenion. Thee gradient imply a non-zero tangential tre bondary ondition (Marangoni effet). 1. INTRODUCTION Srfae tenion i a very important mehanim for mall ale flow h a paint film, the motion of a ontat len on the eyeball or vario wetting or oating flow. For example, to model paint film or foam, it may be important to take rfae tenion gradient into aont, giving rie to a different extra term in evoltion eqation. Sh gradient give rie to the o-alled Marangoni flow, nexpetedly, and they have been fond to dominate many zero-gravity flid dynami experiment arried ot in pae, in partilar thoe onerned with rytal growth. The ability of the rfae tenion to vary patially i alo a rial ingredient for the flid to be able to form a foam. It i alo believed to be the mehanim reponible for the ripple that are often oberved on olvent-baed paint film.. PROBLM FORMULATION Conider the flow of a thin layer of an inompreible Newtonian flid with ontant denity ρ and ontant vioity µ down a horizontal plate. The flow i driven imltaneoly by gravity and a rfae tenion gradient = σ / x. Univerity of Oradea, Department of Mathemati and Informati, 3-5 Armatei Române, 3700 Oradea. Rev. Rom. Si. Teh. Mé. Appl., Tome 54, No., P , Baret, 009

2 8 milia Borşa e hooe Carteian axe Oxyz, with the x axi in the diretion of flow and the z axi tranvere to the diretion of flow (Fig. 1) and o the veloity i given by =(x,z,t)i +w(x,z,t)k. z = σ / x O h(x,t)=z x Fig. 1 The geometry of problem. In the thin film approximation the Navier-Stoke eqation read = ( 1 / ρ) p / x + ν( / z ) + f, (1) = ( 1 / ρ) p / z + f, () where f 1 = 0, f = g. Here z = h(x, t) i the nknown eqation of the freerfae, p i the prere in the flid, g i the gravitational aeleration, µ = ρν i the oeffiient of the dynami vioity, and p depend on x = (x, y, z, t), where t i the time and z point pward. Moreover, the non-lip ondition mt be atified = 0, at z = 0. (3) The normal tre and tangential tre (hear) at the free rfae z = h(x,t) imply the following two bondary ondition p =, at z = h(x, t), (4) p 0 where p 0 i the atmopheri prere, and µ ( / z) =, at z = h(x, t). (5) The ondition (5) i the Levih-Ari bondary ondition. In addition to the dynami bondary ondition, on the free rfae we impoe the kinemati bondary ondition w = h / t + h / x, at z = h(x, t). (6) Integrating (1) nder the tre-free bondary ondition we obtain

3 3 Thin film flow driven by gravity and a rfae tenion gradient 83 = ( ρg / µ ) h / x z + ( / µ ρg / ν h / x h) z, (7) w = ( ρg / 6µ ) h / x z 3 + ( ρg / µ ) z [ h / x h + ( h / x ) ]. (8) Finally, ing the ontinity eqation in the thin-film approximation and the kinemati bondary ondition lead the evoltion eqation for z = h(x, t) ρ g µ h h x x= h + µ h h x. 3 /3 ( / )/ t / / At thi point, it i onvenient to introde apropriate nondimenional variable defined by h = h / h0, x = x / L, t = tu / h0, = ( L ) / σ0, (10) where the veloity U and the length ale L are harateriti qantitie of the problem. Ame that δ = h 0 / L << 1, where h 0 i the harateriti length for the film thikne. Then onvert the eqation (9) into nondimenional form in term of the nondimenional variable h, x, t. For the ake of impler notation we drop the tar. In thi way the eqation (9) beome 3 Bo ( / ) / t Ca / h h x x= h + h h x, 4 where Bo =δ ( ρ gl ) /(3 µ U) i the Bond nmber and Ca = ( Lσ 0 / µ U) δ i the appilarity nmber. For the nonlinear eqation (11), we an find the wave oltion (9) (11) h = Y ( x t), (1) whih Y atifie the ordinary differential eqation 3 '' ' ' ' Bo( YY + 3 YY ) = Y + Ca YY. (13) Here the prime denote the differentiation with repet to T = x t. Let write (13) a the ytem of two ordinary differential eqation ' Y = X, (14) ' 3 X X Y X Y X Y = /Bo / + Ca /Bo / 3 /. (15) It generate a dynamial ytem all eqilibria of whih are nonhyperboli. liminating T between the two eqation (14 15) we obtain the ordinary differential eqation

4 84 milia Borşa 4 the oltion of whih read 3 d /d /Bo 1/ Ca /Bo 1/ 3 / X Y = Y + Y X Y, (16) 3 X ( Y) = /(BoCa ) 1/ Y /(Bo Y ) + Ca /(Bo Y). (17) and it Taylor expanding abot / Ca i The phae portrait i hown in Fig.. XY ( ) (Ca) /(Bo )( Y / Ca ) =. (18) Y o Ca O X Fig. Phae portrait. Here,, are the invariant table, ntable and enter manifold and,, are the orreponding invariant bpae for the linearized ytem. 3. TH SPCIAL CAS Ca = 1 In the ae Ca = 1, the ytem (14 15) rede to Y ' = X, (19)

5 5 Thin film flow driven by gravity and a rfae tenion gradient 85 = /Bo / + /Bo / 3 /. (0) ' 3 X X Y X Y X Y and, imilarly a in Setion, it implie 3 d /d /Bo 1/ /Bo 1/ 3 / X Y = Y + Y X Y, (1) The oltion of thi ordinary differential eqation read 3 X ( Y) = ( /Bo ) 1/ Y /Bo 1/ Y + /(Bo Y), () h that it Taylor expanding abot And the phae portrait i hown in Fig. 3 / i XY ( ) ( /Bo ) ( Y / ) 4 3 =. (1) Y o O X Fig. 3 Phae portrait for Ca = 1. Let remark that the flow with thikne maller then Y = / are table. 4. CONCLUSION In lbriation approximation we have onidered the flow of a thin layer on a horizontal plate. The flow i driven imltaneoly by gravity and ome gradient of rfae tenion. Thee gradient imply a non-zero tangential tre bondary ondition (Marangoni effet).

6 86 milia Borşa 6 Thi tdy how that there are table tationary oltion in the flow of the thin layer. Reeived Marh 15, 005 RFRNCS 1. ACHSON D.J., lementary Flid Dynami, Oxford Univerity Pre, Oxford, United Kingdom, 1990, pp AMICK C.J., FRANKL L.., Steady Soltion of the Navier-Stoke qation Repreenting Plane Flow in Channel of Vario Type, Ata Mathematia, 144, pp (1980). 3.. BORŞA, C.I GHORGHIU, Flow of a Vio Thin Layer on an Inlined Plane Driven by a Srfae Tenion Gradient, Reve d Analye Nmeriqe et de la Theorie de l Approximation,, pp (001). 4. BRAUN, R.J., Mrray, B.T., Lbriation theory for reative preading of a thin drop, Phy. Flid, 7, p (1995). 5. CHIFU,., GHORGHIU, C.I., STAN, I., Srfae Mobility of Srfatant Soltion XI. Nmerial Analyi for the Marangoni and Gravity Flow in a Thin Liqid Layer of Trianglar Setion, Rev. Romaine Chim, 9, 1, pp (1984). 6. DUFFY, B.R., MOFFATT, H.K., Flow of a vio trikle on a lowly varying, The Chemial ngineering Jornal, 60, pp (1995). 7. DUFFY, B.R., and ILSON, S.K., A third-order differential eqation ariing in thin-film flow and relevant to Tanner` Law, Appl. Math. Lett., 10, p. 63 (1997). 8. OCKNDON, H., OCKNDON, J.R., Vio Flow, Cambridge, Univerity Pre, 1995, pp LVICH, V.G., Phyio-Chemial Hydrodynami, nglewood Cliff, New Jerey, TOLL, G.D. and ROTHFLD, L.B., Hydrodynami of rivlet flow, AICH J., 1, p. 97 (1966). 11. ILSON, S.K., DUFFY, B.R., On the gravity-driven draining of a rivlet of vio flid down a lowly varying btrate with variation tranvere to the diretion of flow, Phy. Flid, 10, pp. 13- (1998). 1. HITAKR, S., ffet of Srfae Ative Agent on the Stability of Falling Liqid Film, I and C Fndamental, 3, pp (1964). 13. YOUNG, G.., DAVIS, S.H., Rivlet intabilitie, J. Flid Meh., 176, 1 (1987).